Let a linear operator P be equal to f(P1,...,P_N), where f is a polynomial and P_1,...,P_N are rank 1 operator. We present a procedure, called discrete path integration, to calculate the characteristic polynomial of P. The procedure itself resembles the well-known method in quantum physics; examples of results obtained include the celebrated Matrix-tree theorem (G. Kirchhoff, 1847), some its generalizations, and several results in discrete differential geometry (joint work with A. Ploskonosov and A. Trofimova).